Week 7 Activities Reflection: Nick Sayers Interview

What a treat to see the STEAM powered artist, Nick Sayers’ interview/presentation this week. There were so many great examples of linking math, (science), and art. Here are a few things that I connected with:

Stop 1: Math/art can be cultural and political. In my reading this week (Hart, 2024) it was explained that math/art may not really be embraced as art within the art community because it lacks culture and social/political viewpoints. Well, that is not the case with Nick Sayers work! He showed that math/art can in fact “speak truths.” (Hart, 2024, p. 524) From his formation of spheres (unending surfaces) out of materials like train tickets (garbage) to represent the endless waste that our daily lives produce, to the Hilbert-Moore space filling curve outlined with sandcastles made out of toxic sand from the Aral Sea Desert representing colonization and environmental contamination as we take over land and fill it with toxins. I find the blend of art used to represent and explore mathematics while connecting to real-life issues inspiring and think that it would be very engaging for youth. An important reminder that maths do not have to be separate from culture and/or politics.         

            

Train ticket Sphere (21:20)              Toxic sand castle space filling curve (1:51:05)

Stop 2: “Being an artist is so much more than being able to draw a picture.” (10:01) Sayers says this paralleling the idea that being a mathematician is so much more than being good at arithmetic. What an amazing message to get out there! Just as art can be sculpture, paint, textile, dance and more, so can mathematics be more than multiplication facts and counting change. Both of these subjects come with many thinking “I’m not good at art / math,” and that notion holding them back from exploring and experiencing the disciplines. When we broaden what “counts” as art or as math we open it up to more people to be willing to try it. 

Stop 3: Scaling. Ever since the our 551 class I have been interested in a scaling project with students. (Maybe in the spring with my 9s…?) I really resonate with the ideas of representing large, or small, numbers and quantities and making them more relatable to connect to. Sayers has many great examples of scaling, including his pantograph or “body minimizing machine,” cycling the solar system and even pinhole solargraphs that scale time. I would love to try making a pantograph to make scale drawings in the classroom. I wonder if a shop teacher would be interested in collaborating on that with me. What a real-time way to experience scale. Also, I appreciated that the cycling activity actually traversed a distance from spot to spot. I had not thought of embodying a scaling activity this way, as an outing similar to an orienteering activity.  

Stop 4:  Planning vs go-with-the-flow. Around 45:00 Susan and Nick have a conversation around how much of an activity should be planned and how much should go-with-the-flow as new understandings are developed from the community of people interacting with the task. This conversation is sparked from Nick’s bike gear spirograph that he added 4 pens to within a school setting for kids to try out. It led to different patterns and understandings than 1 pen showed. As teachers, this is a great question. How do we strike the balance between structured tasks and organic developments that come out of the class learning together? On one hand we need to be prepared so that class management stays intact, and that we are linking to curricular outcomes that we want to hit. On the other hand, we want to sink into questions, wonders and ideas that students put forth to promote exploration and engagement. Especially as we bring embodied experiences into the classroom - the whole point here is to play, feel and experience. We want connections to form out of this. The thing is, I have limited time to take ideas that come out of class one day and then plan how to implement them in the classroom in an organic fashion the next day, or even better, have the confidence and knowledge base to implement them on the fly. (I think that timing is important here to keep momentum in the learning and interest.) When I leave work, I have family commitments etc. So how does one strike this balance effectively? Maybe by pre-thinking though what students might connect to? Or maybe it can wait a week until that thinking on my part can happen, and circle back? 

Questions for Nick: 

• When you begin a project (like the train ticket sphere or sandcastles), does the mathematical structure come first or the cultural/political message? Or do they emerge together? Do you ever worry that viewers will focus only on the math and miss the political commentary, or vice versa? In pieces like these, that are not as interactive as something like the bike gear spirograph or cycling the solar system, is there a collaborative portion to getting to your final outcome? Or are they more solo acts?   

References:

Hart, G. W. (2024). What can we say about “math/art”? Notices of the American Mathematical Society, 71(4), 520–525.

Sayers, N. (2026, February 18). Nick Sayers interview [Interview by S. Gerofsky] [Video]. Vimeo. https://vimeo.com/1166172275/3a7a243bce 

Week 7 Reading: What Can We Say About “Math/Art”? (Hart, 2024)

Summary: Hart, an applied mathematician and sculptor, starts his article with two side by side images. One of his sculpture, Solar Flair, that is based on the A5 symmetry of the icosahedron and one by Jasper Johns, Numbers, that shows a 2x5 array of the digits 0-9. With these he poses the question: “Math/art is becoming a standard term, but what exactly is it or should it be?” (p. 520)  

Hart goes on to explain that while the field of math/art is growing, there is not yet a framework that defines or situates the field. What are its core values? Is it a branch of applied mathematics? Or a separate discipline altogether? To answer these questions, he suggests that examples be studied to look for connections and generalizations. 

Hart feels that math/art products must bring forward aspects of “mathematical pleasure.” (p. 524) He argues that math appeals to people because of the joys in reasoning and understanding with clarity, and that mathematical artworks should perpetuate those aspects. He feels that math/art can deepen reasoning and understanding by “somehow saying something beyond words.” (p. 525) Audiences of math/art may include mathematicians, but also math educators, math learners and the general public. It is accessible at many levels, to deepen your mathematical understanding and make new connections. 

Hart acknowledges that traditional art institutions like galleries and museums are not showcasing math/art readily. He figures this may be because products are not traditional “fine art” formats that would showcase culture, but rather are often crafts, designs, models and visualizations. He wonders if attaching a name other than “art” to math/art might make it more accepted by artists and art fields. This opens the discussion to math/art being a space that is maybe not within the art field or the math field, but rather a bridge in between, a space unto itself. 

The article concludes with Hart encouraging everyone, especially mathematicians, to create art. Not only because it is rewarding, but because it is akin to solving a difficult problem. It lifts your spirit and promotes deeper understanding. 

Stop 1: “One cannot hope to approach the topic as in a text book with definitions and theorems already laid out. Instead, one must see it more like a challenge as a group problem-solving session, where one ponders examples and counter-examples and enjoys the communal process of beginner to sort through and make sense of an initially confusing cloud of ideas.” (p 521) This made me think of the mathematical mindsets and habits we want to promote in students. Right now, for most of mine, it is hard to take the time to explore and persist with a task. Getting out of the habit of an instant, spoon-fed answer and into dialogue, I am finding, is a tough transition. Perhaps exploring examples of math/art and making our own math/art can continue to foster this transition.   

Stop 2: “Artists generally aim to communicate something to their viewers. In contemporary fine art, the message is often a social or political viewpoint, with the artist daring to push boundaries and speak truths not otherwise heard. Math/art is characteristically tamer.” (p. 524) A few times throughout the article, Hart notions that math/art is maybe seen as “lesser” art because it lacks culture and impact. I am not sure that I agree. As someone (like many of us) who was raised in traditional-school-type-math experiences, I am just now seeing mathematics as broad and open and creative and with so many more possibilities than I realized. I think that math/art pushes us out of this traditional math culture and helps to expand mathematics to more people. Isn’t that pushing a boundary and speaking a truth? Maybe because Hart is a mathematician that already lives in this broad, open and creative world of maths he doesn’t see it the same, but I would argue that much of the population (and education) is not in that same place.       

Stop 3: “Those who have journeyed through mathematical lands have unique stories to tell of what they found and how they now see the world.” (p. 525) I think this is an absolutely beautiful quote. It is situated in the concluding aspects of the article when Hart is encouraging everyone to make art (especially mathematicians.) I think that this statement is true, not only for mathematicians that want to share the story of their journey of understanding, but to each and every student that brings their unique perspective and approach to looking at, learning and understanding maths. This inspires me to inspire learners to be creators! 

Questions:

1. Have you used any specific example of math/art in your math practice with students? 
2. What kinds of math/art might you be interested in having students create? 

Reference:

Hart, G. W. (2024). What can we say about “math/art”? Notices of the American Mathematical Society, 71(4), 520–525.

Week 6 Activities Reflection

This week I chose to try part of the Scott Kim and Karl Schaffer Making Stars video with my Math 9 class as a warm up, specifically the section where they used their hands to create 2 different 5-pointed stars:

   

We are working on identifying, explaining and creating patterns, so I asked them to watch for the base, repeating unit of the pattern (we called it the peace sign). Inspired by my reading this week, I followed the exploratory outline of Vogelstein et. al (2019) by having students dissect the video clip to see if they could figure out how the performers did the shapes, then reenact the two 5-pointed stars, and finally create their own shape with a different hand base unit. 

It was fun! Right away I noticed aspects of ensemble learning, as everyone in the table group needed to provide a hand in order for the shapes to work. This got even my most reluctant to participate involved. Reenactment was powerful. While they all got the first star shape right away, the second took a few tries, figuring out who had to join who for finger placement. Extended shapes included peace-sign suns, a heart made out of feet, a square made out of bent arms and a flower made out of hands cupped to make petals.

      

References:

Kim, S. & Shaffer, K. (n.d.). Making Stars [Video]. YouTube. https://www.youtube.com/watch?v=dkIm02yYoog

Vogelstein, L., Brady, C., & Hall, R. (2019). Reenacting mathematical concepts found in large-scale dance performance can provide both material and method for ensemble learning. ZDM: The International Journal on Mathematics Education, 51(2), 331–346.

Week 6 Reading: Reenacting mathematical concepts found in large-scale dance performance can provide both material and method for ensemble learning (Vogelstein et al., 2019)

Summary: Vogelstein, Brady, and Hall (2019) promote foraging in public media for performances with potential for exploring mathematics. This allows viewing mathematics within cultural contexts and opportunity to highlight the complexities of the work. They chose the opening ceremonies of the 2016 Rio Olympic Games to work with for their research. In this performance dancers worked in quartets, each holding a corner of a square piece of fabric (about 7’x7’) to create effects. The dance includes walking scale symmetries, rotations, reflections and translations. Here is a link if you would like to see, starting at 12:15. 

In their study, authors invited learners in groups of 4 to dissect the performance by watching it together and analyzing the dancer's moves and resulting effects with the fabric. Then, learners reenact aspects of the performance, physically moving through different combinations to get the desired effects. Finally, learners create their own performance to culminate the activity. All the way through the process, learners are encouraged to describe their experiences mathematically.   

The authors found that reenactment was pivotal to this process in two key ways. One, it developed understanding for what the performers did in the video in a way that just analysis when watching did not. Until learners tried certain moves, they could not be sure their analysis was correct. Second, reenactment also allowed learners to explore what could be done within the “people-plus-prop system” as there were physical limitations here that were not foreseen in the dissection stage. (p. 334) The authors also found the learning quartets had to undergo what they termed ensemble learning, needing to each move a corner of the sheet as an ensemble, or the movements could not be realized.    

Stop 1: Foraging. I really like the idea of foraging media for mathematics connections. I think that this would be highly engaging for students, and agree with the authors that it could be a way to showcase cultures and complexities. Thinking of the current Olympic opening ceremonies, which I tried to stream part of in class, it could have been a great opportunity. Perhaps also the recent Superbowl halftime show which students chatted about the day after. Students could maybe even do their own foraging to bring in examples. It also makes the idea of bridging dance and math more do-able for me as a non-dancer. It feels OK to explore together with the students and not need to be an expert.  

Stop 2: Reenacting as a way to strengthen understanding of processes and also of the limitations provided by reality constraints. One big question I had coming into this course was around re-creating. I wondered if there was as much learning value in re-creating as in creating something new or your own. During our week re-creating a Bridges piece, I answered this question for myself, noting that re-creation was valuable for me in that I had to slow down, think through and really understand the concepts to be able to attempt it myself. This article (and my activity experience this week) furthers my agreement and has expanded it to include the experience of limitations. I did experience the limitations of my person-plus-materials system during Bridges week, I just didn’t recognize it as such. Limitations are an important part of mathematics.

Stop 3: I love the idea of ensemble learning, specifically in comparison to group work. Ensemble learning is different from group work, because learners “need to act together.” (p. 332) In the case of this article, each quartet member needed to move a corner of the fabric. If one did not, the task could not be performed. This eliminates the group work scenario where someone can “participate” without really doing anything, or really understand what is going on. It builds in a whole new level of accountability. I am quite excited by this idea!     

Question: 

1. How can we build in more ensemble learning in the classroom? Can we do it with a task that does not have a prop? 

Reference: 

IOC. (2016, August 5). Rio 2016 Opening Ceremony – Parade of Nations [Video]. YouTube. https://youtu.be/N_qXm9HY9Ro

Vogelstein, L., Brady, C., & Hall, R. (2019). Reenacting mathematical concepts found in large-scale dance performance can provide both material and method for ensemble learning. ZDM: The International Journal on Mathematics Education, 51(2), 331–346.

Week 5 Activities Reflection

I chose to work with Sarah Chase’s “3 against 2” phasing activity this week; I found her video beautiful, simple and complex all at once. I understood how 2x3 takes 6 movements before repeating, but I wanted to take a closer look at even numbers or multiples to better see the patterns there. I did 2 against 2, and then tried 4 against 8. The pattern was too big for me to master, even trying the word trick that Sarah mentioned, so I had the idea of making it into a team activity. One person (my daughter Kaitlyn) moves 4, a second person (me) moves 8, a third person (my daughter Ella) provides the beat to move to, and a fourth (my husband Ryan) could record the number of movements it takes until realigning/repeating. (This could extend to more people joining the phase as well, for example looking at 2 against 3 against 5.)   

Right away, my brain went to looking for patterns with lowest common multiples and factors. When I mapped the process out on a big white board, I could see that the reset was happening at the lowest common multiple.  

I think phasing like this could also work well when learning how to read an analog clock. One student could walk 60 steps in a circle (second hand) and when they get back to the top a second student can take one step forward (minute hand). In this way the phasing becomes, like Sarah says in the video, an abacus that marks and counts tracking relationships. Similarly, one could also use it to look at regrouping and place value with base 10 and other bases.  

Drawing on my reading from Deitiker (2015) this week, using this embodied practice could invite students to explore the process of mathematical relationships first and formalize structured definitions/equations later. I have tried to pose a lesson sketch that would invite students into the story of this experience. 

• When can the friends 3 and 2 meet up? Explain, and have students demonstrate, the phasing activity of 3 against 2 to see how long until the friends can get together. 
• Group students (2 actors, a beat keeper and a recorder) to introduce new friendships. Students can take on the role of acting the number character and test friendships like 3 and 4, 4 and 5, 4 and 8, and 2 and 6. Have groups predict when the friends will meet and design a notation to record their experiences - as Antonsen (2016) said in his video during week one, it is OK for students to make their own notations and definitions while making meaning.  
• Have groups analyze their notations to look for trends and make a conjecture to try out. Can they find a counterexample for their conjecture? For another group’s conjecture? 
• Share conjectures in a whole class discussion and when it is realized that get togethers happen at the first shared number, introduce the formalized term and a mini demo on lowest common multiple.
• Choose your own adventure ending - groups can choose to try using new language and notation for LCM to predict outcomes for when three friends can get together (ex: 2 and 3 and 5) or identify which mystery friends might be meeting based on a set of clues like: they meet on beat 24, and one other time before 30, then justify their choices using the new language.   

Hopefully, by grounding in a shared physical experience this way, learners will have a place for the abstractions to land and view mathematics as a way of describing patterns in the world rather than as a set of disconnected procedures. It flips the lens from starting with “this is a least common multiple and here is how we find it” to “wait…why did that take 6 beats? Will the same pattern happen in this context?” I know the second option would engage me more in the story. 

References: 

Antonsen, R. (2016, December 13). Math is the hidden secret to understanding the world [Video]. TED. YouTube. https://www.youtube.com/watch?v=ZQElzjCsl9o

Chase, S. (2016, November 10). Dancing combinatorics, phases and tides [Video]. Vimeo. https://vimeo.com/251883173

Dietiker, L. (2015). What mathematics education can learn from art: The importance of considering form and experience. Educational Studies in Mathematics, 89(1), 27–44. https://doi.org/10.1007/s10649-015-9592-3

Week 5 reading: What Mathematics Education Can Learn from Art: The Assumptions, Values and Vision of Mathematics Education (Dietiker, 2015)

Summary: Dietiker argues that although mathematics teaching has updated curriculum and standards, the way mathematics is delivered in the classroom largely remains the same as past tradition, where teachers explain, students practice, and meaning is expected to appear. She equates math class to reading an instruction manual: predictable, boring and focused on the end product. Dietiker suggests changing lesson delivery, so that it is more like a novel: an engaging experience that unfolds over time, with a sense of resolution at the end. By intentionally re-structuring lessons from organized lists of outcomes towards the aesthetic of literary art, students are drawn into the process, anticipating next steps, wanting to continue on in a way that resembles not being able to put down a good book. Here are some parallels that she uses between literature and mathematics: 

Characters: mathematical objects - 3, 5

Actions: operations - 3 + 5 (these build the plot)

Plot: sequence of actions that withholds/reveals information building “pull” for the reader (p. 7)

Setting: space that characters are found - symbols on a page vs. tiles on a desk vs. points on a graph

When teachers arrange lessons so that patterns gradually become visible, learners can experience mathematical ideas before formalizing them. In this sense, understanding is not delivered through explanation but constructed through a carefully orchestrated experience. Here is a sample sequence that Dietiker lays out, involving the idea of a fair game: 

1. Read an outline of the presented game and discuss what might happen
2. Play the game with peers and analyze results
3. Play the game against teacher with a conjectured strategy
4. Change the game to make it “fair”
5. Share out the resolution with the class 

Stop 1: “It is not an object’s attribute but the individual’s perception and interaction that is the locus of aesthetic.” (p. 2) This reminds me of the idea that beauty is in the eye of the beholder. It focuses on process rather than product. In previous courses, we have talked about stories for meaning-making in mathematics. The engagement factor was often a link to culture and personal reflection, while meaning formed from articulating ideas. While there is overlap, Dietiker seems to focus more on the story as an engagement tool by design, the way we would get sucked into the plot of a good book or a good show. The meaning-making coming from predicting ahead and wanting to figure out the story. I remember a while back seeing novel study work left in the photocopy room while I was in there with another science colleague (senior physics.) I asked - don’t you sometimes wish you just do a solid novel study? To which his reply was a resounding - nope! But, I did. I think this design strategy might work well for me.    

Stop 2: “Imagining mathematics curriculum as a story opens up the possibility of reimagining the mathematical activities by changing the setting.” (p. 6) I am excited by the idea of setting. Some examples are: a linear function presented in a table vs a coordinate plane, on a calculator, with manipulatives, jumping on a number line. I am hopeful that by experiencing stories across varied settings, we might be able to increase transference of mathematical concepts to more contexts, something that I see students struggle with now. 

Stop 3: “Stories that seem to have no point…or are easily predictable are quickly abandoned.” (p. 9) The idea of diversity in the design of the story is important as an engagement tool and something to keep in mind when developing lessons this way. They cannot be just cookie cutter, formulaic. Dietiker presents a solution to this as different genres of math stories. These can then target different interests or learning goals.    

Questions: 

1. Our learners are so diverse, I wonder how this “novel” format can be universally designed so that it is accessible to all. My comfort zone of adaptations comes from the traditional methods that this paper is pulling away from. What happens when students do not make-meaning from process? How do we best support them? (Perhaps this is where collaboration comes in?) 

Reference: 

Dietiker, L. (2015). What mathematics education can learn from art: The importance of considering form and experience. Educational Studies in Mathematics, 89(1), 27–44. https://doi.org/10.1007/s10649-015-9592-3

Draft project outline and annotated bibliography

Here is the link to my draft outline for the final project and accompanying annotated bibliography. I would like to try using printmaking with grade 9s to notice, analyze, re-create, explain and generalize patterns. 

Image Credit: Pressmaster/Shutterstock.com

Week 4: Activity Reflection

The Bridges selections that I observed this week were from a fashion show (2024). There were many textiles that were sewn and jewelry pieces that were 3-D printed. I chose to explore the mathematical pattern behind this earring and necklace set that was 3-D printed by Hasuna Designs: 

The set displays the apollonian packing mathematical pattern. This pattern is a fractal, starting with three circles tangent to each other, then a smaller circle goes into the triangle-like space in between. There can be many variations of the pattern, depending on the orientation of first three circles as illustrated by Holly (2021):  

  

In trying to recreate this pattern, I tried a few different things. First, I thought I might be able to use coins or buttons but the spaces became too small to fill. Second, I tried to drill with different sized bits through a scrap piece of wood, but it was too difficult to plan the radii that I needed. With a little frustration, I settled on plan number three, which was to take paper strips and tape them into rings. This allowed me the flexibility I needed to create the correct sized circles for each space. The tape warped the circles a bit, but I am happy with the results which are now hanging on my white board in my classroom awaiting curious questions tomorrow (I hope!) 

One theme that resonated with me this week between this activity and Vi Hart’s videos was play. In working with the pattern, I tried many things. Some that did work, some that did not. There was a playfulness about this approach. What happens if I do this? Would this work? It took time, and I feel fairly well acquainted with the apollonian circle packing pattern now. I was wondering if recreating something would lead to as much learning as creating something new. I think it did.     

References: 

Holly, J. E. (2021). What type of Apollonian circle packing will appear? The American Mathematical Monthly, 128(7), 611–629. https://web.colby.edu/janholly/files/2021/11/Holly21pdf.pdf

Week 4 reading: Spinning arms in motion: Exploring mathematics within the art of figure skating (Berezovski et al., 2016)

Summary: Berezovski, Cheng, and Damiano use the art and sport of a figure skating upright spin as a mathematical model for creating math tasks. Using the program, Geometer’s Sketchpad, the authors map the pattern of arm movements during a spin from a bird’s eye view. Two models are presented. Here is a snippet of the author’s Table 1, mapping the first aspect of a spin:

Model 1, designed for middle school, maps only forearm movement, and assumes a stationary upper arm. From this simplified model, math tasks have students calculating scale factors, collecting shoulder-to-hand distance data across varying elbow angles, and using scatterplots to determine a functional curve of best fit. Model 2, designed for secondary students, increases complexity by tracking both upper and lower arm segments in motion. Math tasks for this model include solving triangles at specific timestamps, analyzing the ratios of circular arcs, and investigating the symmetry and interior angles of the pentagon formed when the skater’s hands cross. Math tasks for both models require students to use the Geometer’s Sketchpad software. 

Stop 1: The introduction to the article listed the criteria judges use to evaluate a spin in figure skating, and lists facts about fastest and longest spin from the Guiness Book of World Records. I think that this context could be quite engaging for students, especially if they were going to try some spins themselves (which they did not - more to come on that in later stops.) They could compare their time to the records, or look at aspects of the spin being judged and try to change their body to elicit more of those aspects. 

Stop 2: This model/task was designed using Geometer’s Sketchpad software, and the software is meant to be used by students when they do the task (it is said that the map is dynamic, so maybe it moves?) Unfortunately, the links to the files for the task are no longer working, so I could not look at this aspect. Which may have hindered my understanding of the task, leading to my stop 3… 

Stop 3: The math tasks used with this model are forms of measuring, calculating and graphing using the software. Students did not actually move their bodies. I chose this article because I thought that it would combine the art of sport with math, which I think would be engaging for many students. While I am taking away great aspects (modeling real-world scenarios) I am finding it lacking embodiment. Wouldn’t it be better for students to feel a spin? To move their arms at different angels to see what speeds them up or slows them down? It would be hard to measure this, but maybe filming and measuring angles or distances from their own body images could work? Many sports that students participate in involve mathematical structures such as angles, arcs, and symmetry. Opening up exploration of these aspects for students to experience these maths through their own bodies, and then use technology (video, still frames, measurement tools) to model and analyze what they felt might be more in line with what we are trying to accomplish with this course?

Questions: 

1. I have not used a software program in my math teaching like Geometer’s Sketchpad; the closest I have come is Desmos. Have you used a software successfully in a math context? I wonder if the addition of these technologies adds to, or takes away from, embodiment?  

Reference:

Berezovski, T., Cheng, D., & Damiano, R. (2016). Spinning arms in motion: Exploring mathematics within the art of figure skating. In E. Torrence, B. Torrence, C. Séquin, D. McKenna, K. Fenyvesi, & R. Sarhangi (Eds.), Proceedings of Bridges 2016: Mathematics, Music, Art, Architecture, Education, Culture (pp. 625–628). Tessellations Publishing. http://archive.bridgesmathart.org/2016/bridges2016-625.html